Atwood's Machine with 3 pulleys

[atarr3]

Homework Statement

There are three pulleys in this system. The leftmost pulley is attached to ceiling. It has a string running through it with a mass of m attached. The rightmost pulley is also attached to the ceiling. It has the same string running through it with a mass of 3m attached. In the middle of these two pulleys is a third pulley with a mass 2m attached. The same string from the first two pulleys runs through this pulley too. I need to find the accelerations of the three masses in terms of m

Homework Equations

Net force equations

The Attempt at a Solution

I just want to make sure my net force equations are correct before I apply the conservation of string. For the mass on the left, I have $$\sum F=T - mg = ma_{1}$$ for the mass on the right I have $$\sum F = T - 3mg = 3ma_{3}$$ For the mass in the middle I have $$\sum F = 2T - 2mg = 2ma_{2}$$

 

[atarr3]

Just to clarify this, the pulley in the middle is NOT attached to the ceiling.

 

[conker873]

Did you ever find an answer to this problem? Currently working on it for my current physics course and I'm struggling.

 

[Spinnor]

You are replying to an old post (last post Jan28-10). Does your problem look something like the attached?

 

[rwisz]

where T is the tension in the string and a_{1}, a_{2}, and a_{3} are the accelerations of the three masses respectively.

Yes, your net force equations are correct. The tension in the string will be the same for all three masses, but the masses on the right and in the middle will experience a larger force due to their greater masses. This will result in a smaller acceleration for the mass on the left compared to the other two masses. You can use these equations to solve for the accelerations and then apply the conservation of string to determine the relationship between the accelerations of the three masses in terms of m.